arXiv research

Study the relative volume function on AH manifolds and its applications.

Study of manifolds with nonnegative Ricci curvature and slow relative volume growth.

Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given…

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

New results on relative simplicial volume using bounded acyclicity.

New singularity concept in GR: volume singularities.

Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.

Study calculates arbitrage gains between two markets with limited liquidity.

We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and giv…

This paper optimizes trading strategies to minimize risk and maximize profit while accounting for market uncertainty.

New method uses relative capacities of geodesic balls to determine scalar curvature.

In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of Kähler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local collapsing estimate and can be regarded as an analogue of the Bishop-Gromov volume compa…

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

Study proves Volume Conjecture for Reshetikhin-Turaev invariants.

Grimaldi-Pansu metrics are constructed for manifolds with multiple ends.

In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.

Study shows volumes of knot complements are bounded by linear functions of geodesic periods.

We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…

The paper proves uniqueness of a solution in general relativity.

Let $π$ be a finitely presented group. If h is a non trivial homology class in Hn($π$; Z), a theorem of Gromov (see [Gro83], 6) asserts the existence of regular geometric cycles which represent h, whose relative systolic volume is as close as desired to the systolic volume of h, in which we can control the volume of ba…

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

The paper studies a flow of surfaces in spacetime with a focus on curvature evolution.

Upper bounds for Steklov eigenvalues on manifolds with boundary.

On Kahler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov's relative volume comparison, Bonnet-Meyers theorem, and Yau's gradient estimate for positive harmonic functions. The tool is a Boc…

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L …

Study establishes time functions in Lorentzian spaces without requiring manifold structure.

We study the statistical regularities of opening call auction using the ultra-high-frequency data of 22 liquid stocks traded on the Shenzhen Stock Exchange in 2003. The distribution of the relative price, defined as the relative difference between the order price in opening call auction and the closing price of last tr…

Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

Proves well-posedness for Einstein equations with specific boundary conditions.

Unimodular classification of symmetric matrix map-germs.

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular hom…

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

The paper connects quantum $6j$-symbols to tetrahedra volumes via discrete Fourier transforms.

Extends cohomology theory for infinite volume transformation groups.

Study shows simplicial volume is superadditive under specific conditions.

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this asser…

Gradient flow converges to a minimal convex structure.

Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.

Let N be a manifold (with boundary) of dimension at least 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the simplicial volume. As applications, we extend results of Jungreis and Calegari from clos…

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric pr…

We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hy…

Equity auctions show linear price impact up to a large volume, then non-linear.

Every closed geodesic $γ$ on a surface has a canonically associated knot $\widehatγ$ in the projective unit tangent bundle. We study, for $γ$ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of hom…

We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We describe formal solutions of…

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…

Study proves stability and uniqueness for a specific type of flow.

The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.